Let x0 of type ι → ι be given.
Apply nat_ind with
λ x1 . (∀ x2 . x2 ∈ x1 ⟶ nat_p (x0 x2)) ⟶ nat_p (Pi_nat x0 x1) leaving 2 subgoals.
Assume H0:
∀ x1 . x1 ∈ 0 ⟶ nat_p (x0 x1).
Apply Pi_nat_0 with
x0,
λ x1 x2 . nat_p x2.
The subproof is completed by applying nat_1.
Let x1 of type ι be given.
Apply Pi_nat_S with
x0,
x1,
λ x2 x3 . nat_p x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply mul_nat_p with
Pi_nat x0 x1,
x0 x1 leaving 2 subgoals.
Apply H1.
Let x2 of type ι be given.
Assume H3: x2 ∈ x1.
Apply H2 with
x2.
Apply ordsuccI1 with
x1,
x2.
The subproof is completed by applying H3.
Apply H2 with
x1.
The subproof is completed by applying ordsuccI2 with x1.